Problem: The arithmetic mean of nine numbers is 54. If two numbers $u$ and $v$ are added to the list, the mean of the eleven-member list becomes 66. What is the mean of $u$ and $v$?
Explanation: The original nine numbers, with mean 54, must have a sum of $9 \cdot 54 = 486$. After including $u$ and $v$, we have eleven numbers now, and their sum must be $11 \cdot 66 = 726$. Since the only new numbers in this sum are $u$ and $v$, $u+ v = 726 - 486 = 240$. So, the mean of $u$ and $v$ is $\frac{1}{2}(u+v) = \frac{1}{2}(240) = \boxed{120}$.